where \(X_{e} = \delta ^{1/4} L/\lambda \) . The amplitude equation is a particular case of the Weber differential equation
\(y^{\prime \prime }(z) + \left(\nu +1/2-z^2/4\right)y(z)=0\)  with \(\nu =-1/2\) , whose solution is expressed in terms of the parabolic cylinder function \(y(z) = D_{-1/2}(z) \)  [1]}. Hence, the final perturbation solution for the vertical displacement is given by:

W(x)  -  (x) 
